Asymptotics of the finite-temperature Bessel kernel determinant

In this paper, we study the asymptotics of the Fredholm determinant F σ (x, t) of the finite-temperature deformation of the Bessel kernel. This determinant represents the probability that no particles lie in an interval in the edge scaling limit of the finite-temperature fermion system. The logarithmic derivative of the Fredholm determinant, which we denote by u σ (x, t), satisfies an integrable PDE. We derive the large gap asymptotics of F σ (x, t) at finite temperature. Moreover, we obtain the asymptotics of u σ (x, t) in several different regimes in the (x, t)-plane. In the transition regime, we observe a phase transition which is characterized by solutions of the Jimbo-Miwa-Okamoto σ-form of the second Painlevé equation.